Let \(X\) be a topological space. The Borel \(\sigma\)-algebra of \(X\) is the smallest \(\sigma\)-algebra of subsets of \(X\) that contains every open subset of \(X\), namely the intersection of all such \(\sigma\)-algebras. Its elements are the Borel sets of \(X\).
Since a \(\sigma\)-algebra is closed under complements, countable unions, and countable intersections, every closed set, every $F_\sigma$ set, and every $G_\delta$ set is a Borel set.
What is a Borel set?